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Exponential and Logarithm Functions The Basics If n and m are positive integers... a n = a a a (WeBWoRK: a n or a n ) | {z } n Some identities: Examples: 2 5 = 2 2 2 2 2 The Basics If n and m are

  1. Exponential and Logarithm Functions

  2. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2

  3. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8

  4. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8

  5. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8 (2 3 ) 5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 15

  6. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m ( a n ) m = a n ∗ m Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8 (2 3 ) 5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 15

  7. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m ( a n ) m = a n ∗ m (Notice: a m n means a ( m n ) , since ( a m ) n can be written another way) Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8 (2 3 ) 5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 15 2 3 5 = 2 243 >> (2 3 ) 5 = 2 15

  8. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m ( a n ) m = a n ∗ m (Notice: a m n means a ( m n ) , since ( a m ) n can be written another way) Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8 (2 3 ) 5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 15 2 3 5 = 2 243 >> (2 3 ) 5 = 2 15 2 3 ∗ 5 3 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5) 3

  9. The Basics If n and m are positive integers... a n = a · a · · · · · a (WeBWoRK: a ∧ n or a ∗ ∗ n ) | {z } n Some identities: a n ∗ a m = a n + m ( a n ) m = a n ∗ m (Notice: a m n means a ( m n ) , since ( a m ) n can be written another way) a n ∗ b n = ( a ∗ b ) n Examples: 2 5 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 5 ∗ 2 3 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 8 (2 3 ) 5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 2 15 2 3 5 = 2 243 >> (2 3 ) 5 = 2 15 2 3 ∗ 5 3 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5) 3

  10. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice:

  11. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0

  12. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 .

  13. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 . 2. What is a x if x is negative? a n ∗ a − n = a n − n = a 0 = 1

  14. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 . 2. What is a x if x is negative? a n ∗ a − n = a n − n = a 0 = 1 , so a − n = 1 / ( a n ) .

  15. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 . 2. What is a x if x is negative? a n ∗ a − n = a n − n = a 0 = 1 , so a − n = 1 / ( a n ) . 3. What is a x if x is a fraction? ( a n ) 1 / n = a n ∗ 1 n = a 1 = a

  16. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 . 2. What is a x if x is negative? a n ∗ a − n = a n − n = a 0 = 1 , so a − n = 1 / ( a n ) . 3. What is a x if x is a fraction? √ a ( a n ) 1 / n = a n ∗ 1 n = a 1 = a , so a 1 / n = n √ � m . √ a and a m / n = � a m = n n

  17. Pushing it further... Take for granted: If n and m are positive integers, a n = a · a · · · · · a a n ∗ a m = a n + m , ( a n ) m = a n ∗ m . , | {z } n Notice: 1. What is a 0 ? a n = a n +0 = a n ∗ a 0 , so a 0 = 1 . 2. What is a x if x is negative? a n ∗ a − n = a n − n = a 0 = 1 , so a − n = 1 / ( a n ) . 3. What is a x if x is a fraction? √ a ( a n ) 1 / n = a n ∗ 1 n = a 1 = a , so a 1 / n = n √ � m . √ a and a m / n = � a m = n n � 5 = 2 5 = 32 or 8 5 / 3 = √ √ � 3 √ 32 , 768 = 32 Example: 8 5 / 3 = 8 5 = 3 8 3

  18. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = 1 , 2 , 3 , . . .

  19. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . .

  20. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . .

  21. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  22. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2 and n / 3, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  23. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2, n / 3, . . . , n / 15, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  24. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2, n / 3, . . . , n / 100, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  25. What is a x for all x ? If a > 1: (e.g. a = 2) 8 6 4 2 -3 -2 -1 1 2 3 y = a x

  26. What is a x for all x ? If 0 < a < 1: (e.g. a = 1 2 ) 8 6 4 2 -3 -2 -1 1 2 3 x = . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . .

  27. What is a x for all x ? If 0 < a < 1: (e.g. a = 1 2 ) 8 6 4 2 -3 -2 -1 1 2 3 x = . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . .

  28. What is a x for all x ? If 0 < a < 1: (e.g. a = 1 2 ) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2 , n / 3 , n / 4 , n / 5, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  29. What is a x for all x ? If 0 < a < 1: (e.g. a = 1 2 ) 8 6 4 2 -3 -2 -1 1 2 3 x = n / 2, n / 3, . . . , n / 100, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  30. What is a x for all x ? If 0 < a < 1: (e.g. a = 1 2 ) 8 6 4 2 -3 -2 -1 1 2 3 y = a x

  31. What is a x for all x ? If 0 > a : (e.g. a = − 2) 8 6 4 2 -3 -2 -1 1 2 3 -2 -4 -6 -8 x = . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . .

  32. What is a x for all x ? If 0 > a : (e.g. a = − 2) 8 6 4 2 -3 -2 -1 1 2 3 -2 -4 -6 -8 x = n / 3, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  33. What is a x for all x ? If 0 > a : (e.g. a = − 2) 8 6 4 2 -3 -2 -1 1 2 3 -2 -4 -6 -8 x = n / 3 and n / 2, for n = 0 , ± 1 , ± 2 , ± 3 , . . .

  34. What is a x for all x ? If 0 > a : (e.g. a = − 2) 8 6 4 2 -3 -2 -1 1 2 3 -2 -4 -6 -8 x = n / 2, n / 3, . . . , n / 100, for n = 0 , ± 1 , ± 2 , ± 3 , . . . OH NO!

  35. The function a x : 1 < a : 0 < a < 1: 1 1 D: ( −∞ , ∞ ), R: (0 , ∞ ) D: ( −∞ , ∞ ), R: (0 , ∞ ) a = 1: a = 0: 1 D: ( −∞ , ∞ ), R: { 1 } D: (0 , ∞ ), R: { 0 } Properties: a b ∗ a c = a b + c ( a b ) c = a b ∗ c a − x = 1 / a x a c ∗ b c = ( ab ) c

  36. Our favorite exponential function: Look at how the function is increasing through the point (0 , 1): y = a x : a=3 a=2 a=1.5 a=1.1 a=10

  37. Our favorite exponential function: Look at how the function is increasing through the point (0 , 1): y = a x : a=1.1

  38. Our favorite exponential function: Look at how the function is increasing through the point (0 , 1): y = a x : a=1.5

  39. Our favorite exponential function: Look at how the function is increasing through the point (0 , 1): y = a x : a=2

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